Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors

Abstract

We study how the spectral gap of the normalized Laplacian of a random graph changes when an edge is added to or removed from the graph. There are known examples of graphs where, perhaps counter-intuitively, adding an edge can decrease the spectral gap, a phenomenon that is analogous to Braess's paradox in traffic networks. We show that this is often the case in random graphs in a strong sense. More precisely, we show that for typical instances of Erdi¾?s-Renyi random graphs Gn, p with constant edge density p∈0,1, the addition of a random edge will decrease the spectral gap with positive probability, strictly bounded away from zero. To do this, we prove a new delocalization result for eigenvectors of the Laplacian of Gn, p, which might be of independent interest. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 584-611, 2017